4.1. Fluid velocity and particle volume fraction profiles

Figure 3 presents the mean fluid velocity profiles of the turbulent channel flows. Our findings support previous conclusions that suspended passive spheres decrease velocity across the entire profile even though the blocking ratio (the particle diameter to the channel width) here is larger than that in the previous study (Zhu et al. Reference Zhu, Yu and Shao2018). At a high volume fraction (𝜑₀ = 12.7 %; see figure 3 a), the squirmers exhibit a similar feature to passive spheres in decreasing the velocity profile. However, the extent of this decrease varies among different types of squirmers. These differences may be relevant to the distribution of particles near the wall, as shown in figure 4(a). The peak of the particle volume fraction near the wall indicates the generation of a particle layer, resulting in drag enhancement (Picano, Breugem & Brandt Reference Picano, Breugem and Brandt2015; Costa et al. Reference Costa, Picano, Brandt and Breugem2016). Notably, a higher peak corresponds to a higher density of the particle layer, thus creating slower velocity profiles. Note that these peaks appear at the exact location (y ⁺≈ 42) where there is a sudden weakening tendency in the velocity increase. For Chlamydomonas microalgae swimming in still fluids and near a wall, there is also a notable peak in their time-averaged concentration profile, approximately equal to the sum of the cell radius and the flagellar length (Contino et al. Reference Contino, Lushi, Tuval, Kantsler and Polin2015; Ostapenko et al. Reference Ostapenko, Schwarzendahl, Böddeker, Kreis, Cammann, Mazza and Bäumchen2018). In contrast, the present peak’s distance from the wall indicates the thickness of the particle layer, which is about 1.9 times the squirmer radius from the wall to the body centre. This thickness remains consistent regardless of the self-propulsion modes or the particle volume fraction.

Figure 3.

Fluid mean velocity profiles of squirmer- and particle-laden turbulent channel flows at a/H = 0.125, Re _τ = 180 for (a) 𝜑₀ = 12.7 % and (b) 𝜑₀ = 2.54 %.

Figure 4.

Profiles of the particle volume fraction at a/H = 0.125, Re _τ = 180 for (a) 𝜑₀ = 12.7 % and (b) 𝜑₀ = 2.54 %. The vertical axis is dimensionless by the volume fraction of the squirmers. Pusher1 indicates a grid resolution of 32 across its diameter.

Regarding the profiles of the squirmer volume fraction at Re _s = 5, the peak for the pushers is the largest, followed by that for the neutral squirmers and subsequently for pullers. Consequently, the higher peak results in a slower velocity profile within squirmers. This distribution pattern near the wall (see the peaks) is different from the collective behaviours of multiple squirmers between two walls (without external channel flows), in which pushers and pullers have stronger tendencies to accumulate near the wall compared with neutral squirmers (Li & Ardekani Reference Li and Ardekani2014). The disparity between the two lies in the dominance of distinct mechanisms. In a quiescent fluid, the collective behaviour of particles is dictated by their individual motion and hydrodynamic interactions. Hence, the pullers and pushers tend to accumulate near the wall due to the presence of stresslet (strongest far-field interaction), in contrast to the neutral squirmers lacking this characteristic (Spagnolie & Lauga Reference Spagnolie and Lauga2012). However, in this context, the distribution of squirmers is largely determined by the turbulent channel flows, leading to the intense collisions of spherical particles in the near-wall region (Zhu et al. Reference Zhu, Yu and Shao2018). Compared with passive spheres, squirmers generally result in a further weakening of fluid velocity. However, a notable exception arises with pullers, which instigate a slight augmentation in fluid velocity within the bulk-flow region at Re _s = 5 (y ⁺ ≥ 75; see figure 3 a). On further increasing Re _s to 10, the effect of pullers does not exhibit amplification, as evidenced by a peak in figure 4(a) that is higher than that observed at Re _s = 5.

At a low volume fraction (𝜑₀ = 2.54 %), as shown in figure 3(b), comparable yet diminished effects are discerned in the impeding of velocity profiles by squirmers in contrast to those noted at a high volume fraction. Notably, in comparison with passive spheres, a low volume fraction of pullers (Re_s = 5) results in a slightly enhanced velocity profile across the channel. This observation is consistent with the finding that the peak of the passive particle distribution exceeds that of the pullers (see figure 4 b), contrasting with the trend observed at high volume fraction. The reduction in particle volume fraction (from 𝜑₀ = 12.7 % to 2.54 %) leads to a transition in the distribution profile of particles from bimodal to unimodal. This change can be attributed to variations in hydrodynamic interactions. As shown in figure 4(a), the primary peak of particle distribution, from the vicinity of the wall to the centre of the channel, is associated with particle–wall interaction. In contrast, the secondary peak corresponds to particle–particle interaction. Consequently, as particle volume fraction decreases, the strength of particle–particle interaction diminishes significantly, resulting in the disappearance of the secondary peak, as depicted in figure 4(b).

Compared with the point-particle model demonstrating microswimmers (it is valid for particle dimensions far smaller than the Kolmogorov scale of the flow), our finite-sized squirmers are more inclined towards the wall. Zhang et al. (Reference Zhang, Qiu and Zhao2022) suggested that increasing swimming speed (related to the present Re _s ) results in a slight migration towards the bulk-flow region for spherical microswimmers, and there is no noticeable peak (point-particle model). In contrast, the increase of Re _s from 5to 10 strengthens the aggregate behaviours of the pullers in the near-wall region, leading to the hindrance of the velocity profile (see figures 3 and 4). This discrepancy may be attributed to the near-field flow fields induced by the microswimmers. Li & Ardekani (Reference Li and Ardekani2014) concluded that an individual puller is more easily entrapped near the wall at a higher Re_s due to the presence of stresslet, qualitatively having interpreted our results. For the point-particle model, the microscopic hydrodynamic interactions between particles and the wall have not been fully resolved, and different types of microswimmers are not distinguished. Hence, the collective behaviours are primarily affected by the external turbulent flows.

Figure 5.

Profiles of the fluid viscous stress τ _fV , the fluid Reynolds stress τ _fR , the particle Reynolds stress τ _pR and the particle inner stress τ _pI at φ ₀ = 12.7 % (lines) and 2.54 % (circles). (a) Spheres. (b) Neutral squirmers (β = 0, Re _s = 5). (c) Pushers (β = −5, Re _s = 5). (d) Pullers (β = 5, Re _s = 5 and 10, φ ₀ = 12.7 %). The stresses τ ⁺ are normalized by ρ _f u _τ ².

4.2. Flow drag and turbulent behaviours

To analyse the contributions of flow drag, we decouple the total stress of the channel flows filled with squirmers based on the mean momentum equation (Picano et al. Reference Picano, Breugem and Brandt2015; Zhu et al. Reference Zhu, Yu and Shao2018), and it reads

(4.1) \begin{equation} \underset{\tau _{T}}{\underbrace{\tau _{w}\left(1-\frac{y}{H}\right)}}=\underset{\tau _{fV}}{\underbrace{\varphi _{f}\mu \frac{{\textrm d}\langle u_{xf}\rangle }{{\textrm d}y}}}+\underset{\tau _{pI}}{\underbrace{\varphi _{s}\langle \sigma _{p}\rangle _{xy}}}+\underset{\tau _{fR}}{\underbrace{\varphi _{f}\rho _{f}\langle -u_{xf}^{\prime}u_{yf}^{\prime}\rangle }}+\underset{\tau _{pR}}{\underbrace{\varphi _{s}\rho _{s}\langle -u_{xp}^{\prime}u_{yp}^{\prime}\rangle }} ,\end{equation}

where u _x and u _y are, respectively, the velocity components in the x and y directions; the subscripts f and p represent the fluid phase and solid phase, respectively; σ _p denotes the solid inner stress; and $\varphi _{f}$ and $\varphi _{s}$ are the local volume fractions of fluid and solid, respectively. Note that the angle brackets denote the phase averaging of fluid or solid. The five terms in (4.1) denote the total stress τ _T , the fluid viscous shear stress τ _fV , the particle inner stress τ _pI , the fluid Reynolds stress τ _fR and the particle Reynolds stress τ _pR . Figure 5 presents the contributions of the total stress along the normal direction of the wall. The fluid viscous shear stress τ _fV decreases monotonically from the wall to the channel centre, mirroring the analogous trend observed in viscous dissipation. The main finding in this section is that the presence of different self-propelling modes can significantly affect the particle inner stress τ _pI in the near-wall regions differently, particularly pronounced in scenarios characterized by high particle volume fraction (𝜑₀ = 12.7 %). At 𝜑₀ = 12.7 %, the pushers contribute to the highest peak of τ _pI among the squirmers, followed by that of the neutral squirmers and subsequently that of pullers. This pattern may be attributed to the joint impacts of particles’ accumulation behaviours (volume fraction) and the self-propelling features. Predominantly, the elevated peak of τ _pI correlates with the heightened peak of the particle volume fraction in the near-wall region (see figure 4). Additionally, a faster squirmer results in a larger τ _pI (the steady swimming speeds at Re_s = 5 in descending order are a pusher (β = −5), neutral squirmer (β = 0) and puller (β = 5) (Chisholm et al. Reference Chisholm, Legendre, Lauga and Khair2016); for instance, even though the particle volume fraction of pushers in the bulk flow region is smaller than that of passive spheres, pullers and neutral squirmers, τ _pI of the pusher case is still larger than that of other cases). The particle Reynolds stress τ _pR in flows containing squirmers exhibits a slight increase compared with those involving passive spheres. This can be attributed to implementing the self-propelling velocity boundary condition as (2.7). Significantly, the increase in τ _pI is primarily at the expense of diminishing the fluid Reynolds stress τ _fR . This pattern is also replicated in figure 5(d), and the increase in Re_s (from 5to 10) results in the enhancement of τ _pI accompanied by a reduction in τ _fR .

The conclusion outlined above broadly applies to the low particle volume fraction. At 𝜑₀ = 2.54 %, these stresses are further integrated into figure 6 for a more comprehensive comparison. The variation in fluid viscous stress induced by different squirmers is minimal. In contrast, the particle Reynolds stress τ _pR at a specified Reynolds number (Re _s = 5) demonstrates a positive correlation with the squirmers’ steady speeds (Chisholm et al. Reference Chisholm, Legendre, Lauga and Khair2016). However, two notable findings emerge at a low concentration, distinct from those observed at a high concentration. Firstly, the peak of particle inner stress τ _pI for the puller case is smaller than that of passive spheres at 𝜑₀ = 2.54 % (Re _s = 5), resulting in a greater fluid Reynolds stress τ _fR . This observation could be a key factor contributing to the higher flow profile velocity observed for pullers compared with passive spheres (see figure 3 b). Secondly, at 𝜑₀ = 2.54 %, an increase in the swimming Reynolds number of the pullers (from 5to 10) weakens particle inner stress (enhancing fluid Reynolds stress). Interestingly, this variation with Re _s does not translate into an increase in the velocity profile, in contrast to the findings at 𝜑₀ = 12.7 %. This inconsistency may be attributed to the fact that the increase in Re_s enhances the concentration of pullers near the wall more significantly at 𝜑₀ = 12.7 % than at 2.54 % (see figure 4). Consequently, when increasing Re _s from 5to 10, the puller case with 𝜑₀ = 12.7 % causes a more significant hindrance in the velocity profile, while those with 𝜑₀ = 2.54 % do not.

Figure 6.

Comparison of the fluid stress at φ ₀ = 2.54 %. (a) Viscous stress τ _fV and particle Reynolds stress τ _pR . (b) Fluid Reynolds stress τ _fR and the particle inner stress τ _pI .

To quantify the effects of individual stresses on flow drag, the relationship between the bulk velocity and stresses derived from the mean momentum equation (4.1) reads

(4.2) \begin{equation} \frac{{\textrm d}\langle u_{xf}\rangle }{{\textrm d}y}=\frac{1}{\mu \varphi _{f}}(\tau _{T}-\tau _{fR}-\tau _{pI}-\tau _{pR}) .\end{equation}

Table 2.

Contributions of the individual stresses to the flow drag for different squirmers (φ ₀₁ = 12.7 %, φ ₀₂ = 2.54 %).

The bulk velocity of the entire particle suspension flows can be calculated by integrating (4.2) twice:

(4.3) \begin{align} u_{b}&=\frac{1}{H}\int _{0}^{H}(\varphi _{f}\langle u_{f}\rangle +\varphi _{s}\langle u_{s}\rangle ){\textrm{d}}y \nonumber\\ &=\frac{1}{H}\int _{0}^{H}\varphi _{f}\int _{0}^{y}\frac{1}{\mu \varphi _{f}}(\tau _{T}-\tau _{fR}-\tau _{pI}-\tau _{pR})\;{\textrm{d}}\xi \textrm{dy}+\frac{1}{H}\int _{0}^{H}(\varphi _{s}\left\langle u_{s}\right\rangle ){\textrm{d}}y. \end{align}

Equation (4.3) can be normalized by the characteristic length H and the characteristic velocity $u_{\tau }$ , and we follow the definition of Zhu et al. (Reference Zhu, Yu and Shao2018) with the dimensionless terms of the total stress, the fluid Reynolds stress, the particle inner stress, the particle Reynolds stress and the particle mean velocity as C _T , C _fR , C _pI , C _pR and C _up , respectively. The dimensionless values are summarized in table 2. Note that the total stress term in the single-phase scenario is related to the friction Reynolds number, expressed as C _T = Re_τ /3 (Yu et al. Reference Yu, Lin, Shao and Wang2017). At 𝜑₀ = 12.7 %, the total stress C _T for flows containing passive spheres tends to be slightly lower than that for single-phase flows because the fluid volume fraction is less than unity in particle-laden flows. This trend holds in most cases involving squirmers, except pushers. The scenario for pushers deviates from this pattern at a specified particle volume fraction, exhibiting the smallest u _b and the highest total stress C _T , as indicated in table 2. These findings can be attributed to the self-propelling velocities of the squirmers, which introduces additional kinetic energy into the system and consequently alters the power dynamics of the channel flows and enhances the flow drag. This effect is particularly noteworthy for pushers, as they swim at faster speeds than their counterpart squirmers in the presence of fluid inertia (Chisholm et al. Reference Chisholm, Legendre, Lauga and Khair2016). Notably, a larger C _pI corresponds to larger values of C _T and C _pR , but a smaller value of C _fR , thereby quantitatively reproducing the analysis depicted in figure 5. It is observed that the fluid Reynolds stress C _fR for pullers (β = 5) exceeds that for the counterpart pushers (β = −5) at Re _s = 5. This may be attributed to the magnitudes of vorticity around the body, which serve to strengthen turbulent flows. Specifically, an inertial puller can generate more vorticity than its counterpart pusher (Chisholm et al. Reference Chisholm, Legendre, Lauga and Khair2016). At 𝜑₀ = 2.54 %, the results presented in the table provide further quantification of the analysis conducted in the previous section. One of the most significant conclusions confirmed is that pullers demonstrate a drag-reducing effect compared with passive spheres at Re _s = 5. With an increase in Re _s , although the total Reynolds stress (C _fR + C _pR ) is enhanced, the average velocity of both pullers and fluids decreases.

Figure 7.

Profiles of the RMS velocity and Reynolds shear stress for the flow phase (φ ₀ = 12.7 %): (a) u _xf ; (b) u _yf ; (c) u _zf ; (d) $-\langle u_{xf}^{\prime}u_{yf}^{\prime}\rangle$ .

As depicted in figure 7(a–c), our findings corroborate the conclusion that the introduction of passive spheres leads to a decrease in the peak of the streamwise (x) RMS velocity, while simultaneously increasing the wall-normal (y) and spanwise (z) RMS velocities (Shao, Wu & Yu Reference Shao, Wu and Yu2012; Picano et al. Reference Picano, Breugem and Brandt2015). At a volume fraction of 𝜑₀ = 12.7 %, most squirmers cause a decrease in the streamwise RMS velocity compared with passive spheres. However, neutral squirmers lead to a slightly higher streamwise RMS velocity than the passive particle case in the vicinity of y ⁺ ≈ 42. The mechanism underlying this observation may be attributed to the higher volume fraction of neutral squirmers compared with passive spheres (see figure 4), along with the subtle effect of their self-propelling feature. Throughout the flow profile, the self-propelling mode of squirmers emerges as another critical factor that competes with volume fraction to influence the streamwise RMS velocity. In the near-wall region (e.g. y ⁺ ≤ 30), squirmer volume fraction predominates, establishing a negative (positive) correlation between the volume fraction and the streamwise (wall-normal and spanwise) RMS velocity at Re _s = 5– higher volume fractions correspond to smaller (larger) velocities. This pattern mirrors what is observed for passive spheres. Conversely, the self-propelling mode prevails in the bulk-flow region (e.g., y ⁺ ≥ 120). For instance, despite pushers having a smaller volume fraction than pullers, the streamwise RMS velocity for pushers is smaller than that for pullers. For the puller cases, it is intriguing that an increase in Reynolds number (from 5to 10) results in a decrease in the streamwise RMS velocity but an increase in the wall-normal and spanwise RMS velocities across the entirety of the flow profile. Shao, Wu & Yu (Reference Shao, Wu and Yu2012) suggested that passive spheres can weaken large-scale flow vortices in the streamwise direction and strengthen small-scale vortices when interacting with the wall. Hence, our observation may be attributed to the larger particle inner stress (see figure 5 d) for pullers at Re _s = 10, which weakens the large-scale vorticity in the streamwise direction to a greater extent compared with that at Re _s = 5. Meanwhile, pullers with larger fluid inertia (e.g. Re _s = 10) can induce more small-scale vorticity around the body in all directions than those with small ones (e.g. Re _s = 5) (Chisholm et al. Reference Chisholm, Legendre, Lauga and Khair2016). Since the (small-scale) large-scale vorticity dominates the turbulence in the streamwise direction (wall-normal and spanwise directions), the total turbulence intensity is weakened (enhanced). Comparing figures 7(a) and 7(d), the disparities in the fluid Reynolds stresses for different squirmers exhibit notable consistency with those of the streamwise RMS velocities, thus affirming the dominance of turbulence intensity in the streamwise direction, as mentioned above. The RMS velocities and Reynolds stress for the particle phase are shown in figure 8. Its pattern differs from that of the fluid phase. This discrepancy arises because the behaviour of the fluid phase is primarily influenced by the near-wall vortex structure, while particles are closely associated with their own motion (Zhu et al. Reference Zhu, Yu and Shao2018). It is worth noting that at the wall, the RMS velocities do not vanish due to collisions with the wall, and different values for different squirmers indicate varying collision frequencies.

Figure 8.

Profiles of the RMS velocity and Reynolds shear stress for the particle phase (φ ₀ = 12.7 %): (a) u _xp ; (b) u _yp ; (c) u _zp ; (d) $-\langle u_{xp}^{\prime}u_{yp}^{\prime}\rangle$ .

Figure 9.

Mean direction cosines of the squirmers’ symmetry axes (φ ₀ = 12.7 %). (a) Streamwise component (|cos α|). (b) Wall-normal component (|cos β ₀|).

4.3. Squirmer orientation, rotation and clustering behaviours

Figure 10.

The PDFs of the squirmers’ orientation. (a,c,e) In the bulk-flow region of 135 ≤ y ⁺ ≤ 180 (φ ₀ = 12.7 %). (b,d,f) In the near-wall region of y ⁺ ≤ 45 (φ ₀ = 12.7 %). Here cos α, cos β ₀ and cos γ denote the direction cosines in streamwise, wall-normal and spanwise components, respectively.

Similar to the analysis of axisymmetric passive spheres, the orientation of a squirmer is described using the direction cosines (|cos α|, |cos β ₀|, |cos γ|), where α, β ₀ and γ, respectively, denote the angles between the particle symmetry axis and the streamwise, wall-normal and spanwise directions. Figure 9 presents the mean direction cosines of the squirmers’ symmetry axes. In this section, the algorithm of orientation dynamics for nonspherical particles is also employed to handle passive spherical particles (Zhu et al. Reference Zhu, Yu and Shao2018). Initially, the body coordinates of the squirmers and passive spheres are defined, and their orientation is tracked accordingly. It is observed that all average values of the passive spheres are approximately 0.5, indicating the isotropy of the spherical particles and confirming the suitability of this orientation definition for the squirmers. Since each finite-size particle orientation can only be counted once at a specified time (located at its centre), no particle statistics are available near the wall (0 < y ⁺ < a ⁺ = 22.5). It is observed that the orientation of the squirmers near the wall is not significantly inclined towards the flow direction. As one moves towards the bulk-flow region, pushers and neutrals tend to align their axes more along the flow direction, with pushers showing a more pronounced change. In contrast, pullers tend to favour alignment with the normal direction slightly. Our findings significantly differ from those reported in Zhang et al. (Reference Zhang, Qiu and Zhao2022), where the orientation of microswimmers with an aspect ratio of unity only exhibits minor alterations near the wall and displays isotropy in other regions. This discrepancy may be attributed to the use of a point-particle model, which is suitable for a lower particle volume fraction and does not account for the fluid inertia of the microswimmers and particle–particle interactions. The axisymmetric characteristics of finite-size squirmers and interactions between squirmers and the wall are likely the primary mechanisms contributing to our observed results. The inclusion of self-propelling velocity breaks the geometric isotropy of spherical particles, causing squirmers, especially pushers with lower volume fractions in the bulk-flow region, to adopt specific orientations induced by vorticity in the streamwise direction. Inertial pushers facilitate vorticity convection along their rear (Chisholm et al. Reference Chisholm, Legendre, Lauga and Khair2016; More & Ardekani Reference More and Ardekani2020), enhancing their axial orientation characteristics and making them similar to prolate spheroids. Conversely, inertial pullers engender the accumulation of vortices on both sides of their symmetry axis (Chisholm et al. Reference Chisholm, Legendre, Lauga and Khair2016; More & Ardekani Reference More and Ardekani2020), thereby weakening their axial orientation characteristics and making them similar to oblate spheres. Consequently, the orientation of squirmers in the bulk-flow region aligns with the prevailing notion: particles orient their long axis along the direction of the local fluid vorticity axis (Zhu et al. Reference Zhu, Yu and Shao2018). However, the scenario differs markedly in the near-wall region. Elevated volume fractions of squirmers engender robust hydrodynamic interactions and collisions between particles and with the wall, mitigating the anisotropic properties induced by self-propelling velocity. Near the wall, squirmers tend to exhibit an isotropic orientation distribution akin to that of spherical passive spheres (Zhu et al. Reference Zhu, Yu and Shao2018).

Figures 10(a), 10(c) and 10(e) depict the probability density function (PDF) of the squirmer orientation within the bulk-flow region (135 ≤ y ⁺ ≤ 180). The PDF tends towards unity in all directions for passive spheres, further confirming their orientation isotropy. In the bulk-flow region, as shown in figure 10(a), the PDF peaks of pusher orientation occur at cos α = 1 and −1, indicating alignment predominantly parallel to the streamwise direction, consistent with the results shown in figure 9(a). Additionally, the peak at cos α = 1 is higher than that at cos α = −1, suggesting a significantly higher probability of orienting in the positive streamwise direction compared with the negative direction. This orientation may contribute to the reduction in mean flow velocity in the flow direction by pushers (as referenced in figure 3), as their induced flows (pushing for self-propulsion) oppose the streamwise direction. For neutrals, their orientation characteristics in the streamwise direction are less pronounced compared with pushers, showing a monotonic increase across the entire range of −1 ≤ cos α ≤ 1. Interestingly, for the pullers at different Re _s , there is a peak at cos α ≈ 0.37 in the streamwise direction, corresponding to an angle of α ≈ 68°. In figures 10(c) and 10(e), within the bulk-flow region, the orientation of squirmers appears somewhat similar in the wall-normal and spanwise directions. However, the notable difference lies in the deviation of the wall-normal cosine peaks for neutrals and pushers from cos β ₀ = 0, with deviations of approximately 3° and 5°, respectively. Both neutrals and pushers tend to align perpendicularly to the wall-normal direction, although the pushers show a more pronounced inclination. The cosine orientation values in the wall-normal and spanwise directions for the pullers exhibit minor variation. Two peaks are observed: one parallel to the wall-normal direction (β ₀ = 180°) and another at β ₀ ≈ 27°. The orientation cosine of the squirmers is symmetric along cos γ = 0, indicating uniformity in fluid alignment in the spanwise direction.

For comparison, figures 10(b), 10(d) and 10(f) depict the PDF of the squirmer orientation within the near-wall region (y ⁺ ≤ 45). Unlike pullers or pushers, neutrals do not generate vorticity, making them less likely to be trapped near the wall (Li & Ardekani Reference Li and Ardekani2014). Consequently, the orientation of neutrals does show a pronounced tendency in the streamwise direction. As shown in figure 10(b), there are peaks indicating tendency angles, with α ≈ 35° for pushers and α ≈ 128° for pullers at Re _s = 10. In the wall-normal direction (see figure 10 d), pushers and neutrals exhibit peaks at cos β ₀ = −1, corresponding to an angle of β ₀ = −180°. This indicates that pushers and neutrals tend to swim towards the wall more than pullers, leading to larger aggregations near the wall (see figure 4). Figure 10(f) shows that pushers tend to be perpendicular to the spanwise direction, whereas pullers tend to be parallel to it.

Figure 11.

Squirmer mean rotational rates (φ ₀ = 12.7 %) in (a) tumbling component and (b) spinning component.

In this section, the rotational behaviour of the squirmer is analysed in terms of ‘spinning’ and ‘tumbling’ actions, with spinning denoting rotation around its symmetry axis ( $\langle w_{z}w_{z}\rangle$ ), and tumbling referring to rotation around the other two axes ( $0.5(\langle w_{x}w_{x}\rangle +\langle w_{y}w_{y}\rangle )$ ). Figure 11 illustrates the variations in spinning and tumbling speeds across different squirmers, spanning from positions near the wall to the channel centre. Indeed, the rotational speed profiles of spherical passive spheres demonstrate striking similarity, emphasizing their geometric isotropy. The spinning and tumbling velocities of neutral squirmers closely resemble those of passive spheres within the flow field away from the wall, likely attributable to the lack of self-induced vortices in neutral squirmers, similar to passive spheres. Conversely, the vortices generated by pullers and pushers interact with the flow vortices within the channel, leading to pushers displaying increased (decreased) spinning and reduced (enhanced) tumbling speeds. This discrepancy primarily stems from pushers’ inclination to transport their induced vortices downstream, thereby elongating the effective geometric attributes of particles along their symmetry axis, whereas pullers tend to gather vortices around themselves, elongating the effective geometric properties along the other two axes (Chisholm et al. Reference Chisholm, Legendre, Lauga and Khair2016; More & Ardekani Reference More and Ardekani2020). Zhao et al. (Reference Zhao, Challabotla, Andersson and Variano2015) suggested that ellipsoidal particles exhibiting heightened spinning speeds and diminished tumbling speeds can be explained by their tendency to align their long axes with the direction of the channel flow’s vorticity. The facilitation of spinning and tumbling for pushers and pullers, respectively, arises from the alignment of their elongated axes (the symmetry axis for pushers and the transverse axes for pullers) with the direction of the channel flow’s vorticity. As the Reynolds number increases, pullers can induce more vortices around them, leading to enhanced tumbling speeds. Notably, certain squirmers near the wall display minor peaks in rotational speed, likely due to the complex interactions between squirmers and the wall, as well as robust near-field hydrodynamic interactions.

Without considering external flows, previous studies have shown that squirmers can generate large-scale flow structures, thereby inducing aggregation motion (Li & Ardekani Reference Li and Ardekani2014; Lin & Gao Reference Lin and Gao2019; Qi et al. Reference Qi, Westphal, Gompper and Winkler2022; Zhang et al. Reference Zhang, Yu, Ouyang, Xia and Lin2024). Hence, we consider the pair-correlation function g(r), representing the probability of finding a squirmer at a centre-to-centre distance r from another one, to capture the topological structures of squirmer clusters in the turbulent channel flows, and it reads

(4.4)

(4.5) $\delta (r_{m,n}-r)=\left\{\begin{array}{l} 1,\quad r-\frac{{\unicode[Arial]{x0394}} r}{2}\leq r_{m,n}\lt r+\frac{{\unicode[Arial]{x0394}} r}{2}\\ 0,\quad \text{otherwise}, \end{array}\right. \qquad\quad$

where r _m,n is the distance between squirmers m and n, and V (r _m,n ) = (4π/3)[(r _m,n + Δr/2)³ − (r _m,n − Δr/2)³] is the volume of the spherical shell of radius r _m,n and thickness Δr. As shown in figure 12, the pair-correlation functions of squirmers exhibit a first peak around r = 2, reaching a maximum value before rapidly decreasing. This indicates close contact due to pairwise interactions, with the highest peak signifying the most intense interactions between particles. This result correlates with the particle aggregation observed in figure 4(a), where pushers and passive spheres display the highest and lowest peaks near the wall, respectively. A second peak appears around r = 4, consistent with the reported series of double-peak structures at r = 2, 4, 6, 8,… for swimmer clusters (Ishikawa et al. Reference Ishikawa, Locsei and Pedley2008; Lin & Gao Reference Lin and Gao2019; Zhang et al. Reference Zhang, Yu, Ouyang, Xia and Lin2024). Notably, we did not find any double peaks greater than 4, because of our limited channel space. Interestingly, the second peak for the pushers is smaller than that for the passive spheres, in contrast to the situation of the first peak. This may be because, in a limited channel flow, particles with strong pairwise interactions tend to break the formation of larger particle clusters.

Figure 12.

Pair-correlation function of squirmers in the turbulent channel flows (φ ₀ = 12.7 %).

4.4. Vortex structure

This section investigates the impact of squirmers on vortex structures, aiming to elucidate the potential mechanisms underlying the observed effects of particles on the discussed flow statistics. Figure 13 displays the typical vortex structures in a quarter of the entire calculation domain of the channel flow. We utilize the Q criterion (Q = 1.5) proposed by Hunt, Wray & Moin (Reference Hunt, Wray and Moin1988) for vortex identification, defined as Q = 0.5 ( $\Omega$ _ij $\Omega$ _ij − S _ij S _ij ), where $\Omega$ _ij = (u _ij − u _ji )/2 and S _ij = (u _ij + u _ji )/2 are, respectively, the rotation rate tensor and the strain rate tensor. Comparing figures 13(a) and 13(c), the disparity in vorticity is not pronounced. This can be attributed to the fact that neutral squirmers do not inherently generate vortices; rather, the vortices are primarily induced by hydrodynamic interactions and collisions between particles and the wall. Pushers and pullers both generate vorticity themselves. Pushers contribute to the vortex convection opposite to their swimming direction, whereas pullers tend to attract vortices around themselves. As a result, in the bulk-flow region, noticeably higher vorticity induced by pullers compared with pushers can be observed (the abundance of pullers relative to pushers in this region also plays an important role). Meanwhile, a large number of pushers congregate near the wall, giving rise to numerous small-scale vortices (see figure 13 e). These small vortices are generated both by pushers and by the weakening and fragmentation of large-scale vortices, resulting in a minimal flow RMS velocity in the near-wall region compared with that for other squirmers (see figure 8 a). The orientation and distribution characteristics of the particle phase are depicted in figure 13(b,d,f,h). Upon comparison of figures 13(d), 13(f) and 13(h), it is confirmed that the vorticity in the central region of the flow field is primarily induced by the squirmers themselves: the vorticity generated by neutrals is minimal, whereas that generated by pushers is largely concentrated at their tails, accompanied by no discernible pattern (appearing uniform) in the distribution of vorticity around pullers.

Figure 13.

Vortex structures of the particle-laden channel flows (φ ₀ = 12.7 %) filled with (a,b) passive spheres, (c,d) neutral squirmers, (e,f) pushers and (g,h) pullers. (a,c,e,g) Oblique view with removing particles. (b,d,f,h) Side view in the spanwise direction with particles. The red caps on the squirmers indicate their heads (orientation). A quarter of the calculation domain is displayed to enhance clarity when observing the vortex structure and particle orientation. Note that the colour of the vortices represents the fluid streamwise velocity, and darker means a larger streamwise velocity.